Problem Solving and Computing (CSC-103 98S)

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# Problem Set 7: Conjecturing, Games, and Puzzles

Assigned: Tuesday, April 7, 1998
Due: Tuesday, April 14, 1998

Turn in resolutions for problems 4, 6, and the "Grandpa Polson" problem.

You are encouraged to work on the problems in any order.

1. Painted Tires

[From Thinking Mathematically p. 65. Modified by Sam Rebelsky.]

Once while riding on my bicycle along a path, I crossed a strip of wet paint about six inches wide. After riding a short time in a straight line I looked back at the marks on the pavement left by the wet paint picked up on my tires. What did I see? In particular, did I see separate marks from the front and back tires? What else do you need to know to answer this question?

2. Funny Furniture

[From Thinking Mathematically, p. 67. Modified by Sam Rebelsky.]

A very heavy armchair needs to be moved. It is sufficiently heavy that it is only possible to move it by rotating it ninety degrees about one of its corners. Can it be moved so that it is exactly beside its starting position (i.e., that the left front leg is where the right front leg was and the left rear leg is where the right rear leg was) and facing the same way? If so, describe how. If not, suggest why not.

3. A Family Reunion

[The problem was taken from `http://public-library.calgary.ab.ca/srg/gamesw8.htm` . That site indicates that it was taken from Paul Sloane's Lateral Thinking Puzzlers.]

At a family reunion, it was found that the following relationships existed between the people present: Father, Mother, Son, Daughter, Uncle, Aunt, Brother, Sister, Cousin, Niece and Nephew. However, there were only four people present. How can this be so?

4. Making Fifteen

[From Thinking Mathematically, p. 79. Modified by Sam Rebelsky.]

Nine counters marked with the digits 1 to 9 are placed on the table. Two players alternately take one counter from the table. The winner is the first player to obtain a set of counters that include three counters which sum to fifteen. Devise a winning strategy.

Is there a winning strategy if there are three players?

Are there interesting variants of the game with more counters (and perhaps a different goal number)?

5. Shuffling Coins

[From Java Structures by Duane Bailey. Modified (rewritten) by Sam Rebelsky.]

Professor Perplexed has placed four coins in a five-grid space in order of decreasing diameter. He wants to know how to rearrange the coins so that they are in order of increasing diameter. He's decided to restrict movement so that it is only possible to place smaller coins on top of larger coins (or on top of empty spaces). What is the quickest strategy you can devise to reverse the order of coins?

Your goal is to go from

```+---+---+---+---+---+
| Q | N | P | D |   |
+---+---+---+---+---+
```

to
```+---+---+---+---+---+
|   | D | P | N | Q |
+---+---+---+---+---+
```

6. The Eight Puzzle

[A popular puzzle, described by Sam Rebelsky.]

The eight puzzle is a puzzle played on a 3 by 3 grid. The numbers from one to eight are placed on the grid with one empty space. The goal of the puzzle is to place them in order, as in

```+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 |   |
+---+---+---+
```

You can shift any horizontal or vertical neighbor into the empty space.

Give the instructions for solving the eight puzzle with each of the following initial configurations.

```+---+---+---+
| 2 | 6 | 4 |
+---+---+---+
| 7 |   | 3 |
+---+---+---+
| 5 | 8 | 1 |
+---+---+---+
```

```+---+---+---+
| 5 | 4 | 2 |
+---+---+---+
| 8 |   | 3 |
+---+---+---+
| 1 | 7 | 6 |
+---+---+---+
```

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